There is no Weil Zeta Function per se, but a bunch of such associated with various algebraic-geometric objects (the simplest ones are associated with elliptic curve structures on toruses); such functions are rational functions (very unlike the highly transcendent classical zeta) and the analogue of the Riemann Hypothesis there is a statement about the decomposition in factors of the numerators of such and it's an algebraic-combinatorial statement essentially, not an analytic one like in the classical case.

The RZ book by Patterson explains how the same combinatorial statement extrapolated for the complex field is simply false but that has nothing really to do with the Riemann Hypothesis per se.